The superconductivity in antiperovskite compound MgCNi3 was discovered in 2001 following the discovery of the superconducting MgB2. In spite of its lower superconducting transition temperature (8 K) than MgB2 (39 K), MgCNi3 has attracted considerable attention due to its high content of magnetic element Ni and the cubic structure analogous to the perovskite cuprates. After years of extensive investigations both theoretically and experimentally, however, it is still not clear whether the mechanism for superconductivity is conventional or not. The central issue is if and how the ferromagnetic spin fluctuations contribute to the cooper paring. Recently, the experimental results on the single crystals firstly reported in 2007 trend to indicate a conventional s-wave mechanism. Meanwhile many compounds neighboring to MgCNi3 were synthesized and the physical properties were investigated, which enriches the physics of the Ni-based antiperovskite compounds and help understand the superconductivity in MgCNi3. In this paper, we summarize the research progress in these two aspects. Moreover, a universal phase diagram of these compounds is presented, which suggests a phonon-mediated mechanism for the superconductivity, as well as a clue for searching new superconductors with the antiperovskite structure. Finally, a few possible scopes for future research are proposed.